Identification Of Time-invariant Systems Research Articles

Parsimonious system identification from fragmented quantised measurements

Quantisation is the process of mapping an input signal from an infinite continuous set to a countable set with a finite number of elements. It is a non-linear irreversible process, which makes the traditional methods of system identification no longer applicable. In this work, we propose a method for parsimonious linear time invariant system identification when only quantised observations, discerned from noisy data, are available. More formally, given a priori information on the system, represented by a compact set containing the poles of the system, and quantised realizations, our algorithm aims at identifying the least order system that is compatible with the available information. The proposed approach takes also into account that the available data can be subject to fragmentation. Our proposed algorithm relies on an ADMM approach to solve a quasi-norm objective problem. Numerical results highlight the performance of the proposed approach when compared to the minimisation in terms of the sparsity of the induced solution.

Simultaneous Identification of Time-Varying Parameters and External Loads Based on Extended Kalman Filter: Approach and Validation

Online identification of time-variant parameters without knowledge of external loads is an important but challenging task for structural health monitoring and vibration control. In this study, a two-stage approach, named extended Kalman filter with forgetting factor matrix under unknown inputs (EKF-FFM-UI), is proposed for simultaneously identifying the time-variant parameters and external loads. In stage 1, an extended Kalman filter under unknown inputs (EKF-UI) approach previously proposed by the authors is employed for estimating the structural states and unknown loads. This EKF-UI approach is solely suitable for time-invariant system identification. Therefore, the aim of stage 2 is to improve this approach for the purpose of possessing tracking capability. In this stage, the acceleration responses are first reconstructed by using the differential equation of motion and employed for improving the accuracy of estimated structural states. A forgetting factor matrix is introduced into the priori estimation error covariance matrix to track time-varying parameters. The square errors between the measurements and the corresponding estimates are defined as an index and used for detecting the damage time instant. Then, a covariance resetting technique is employed to assure that such changes in structural parameters can be efficiently captured. A shear-type building structure without/with magneto-rheological (MR) dampers and a fixed beam structure are used as numerical examples for validating the effectiveness of the proposed approach. Experimental tests on a six-story building model are also conducted. Results show the time-varying parameters and unknown inputs can be simultaneously identified with acceptable accuracy.

A recursive identification algorithm for linear discrete periodic systems

The parameter identification problem of linear discrete period time-varying (LDPV) systems is studied in this paper and a two-step identification algorithm based on the recursive least squares principle is proposed. The recursive principle-based identification algorithm is mostly used to deal with the parameter identification of time-invariant systems. Combine with the cyclic reconstruction method, the LDPV system is converted to a linear time-invariant system. Then, based on this and combined with recursive least squares identification, a discrimination algorithm is proposed for the parameter estimation of the LDPV system. Finally, a numerical simulation example is given to demonstrate the effectiveness of the proposed algorithm.

A Randomized Algorithm for Parsimonious Model Identification

Identifying parsimonious models is generically a “hard” nonconvex problem. Available approaches typically rely on relaxations such as Group Lasso or nuclear norm minimization. Moreover, incorporating stability and model order constraints into the formalism in such methods entails a substantial increase in computational complexity. Motivated by these challenges, in this paper we present algorithms for parsimonious linear time invariant system identification aimed at identifying low-complexity models which i) incorporate a priori knowledge on the system (e.g., stability), ii) allow for data with missing/nonuniform measurements, and iii) are able to use data obtained from several runs of the system with different unknown initial conditions. The randomized algorithms proposed are based on the concept of atomic norm and provide a numerically efficient way to identify sparse models from large amounts of noisy data.

Identification of High-Tech Motion Systems: An Active Vibration Isolation Benchmark

The benchmark Active Vibration Isolation System (AVIS) in this paper is a complex high-tech industrial system used in vibration and motion control applications. The system is complex in the sense of high order exible dynamics and multiple inputs and outputs. The aim of this benchmark is to compare difierent black box, linear time invariant system identification algorithms. Difierent large data sets are provided, enabling the use of both frequency domain and time domain identification approaches. The idea of the benchmark is to investigate both model accuracy and numerical reliability of the computational steps. Several reference solutions are provided, which demonstrate the challenging aspects of this benchmark already in the singleinput single-output case. The benchmark data and additional info are available on the website of Tom Oomen.

Lifting approach to simplify output-only continuous-scan laser vibrometry

Continuous-scan laser Doppler vibrometry (CSLDV) can greatly accelerate modal testing by continuously sweeping the measuring laser over a structure, effectively capturing its response at tens or even hundreds of points along the laser path. Several techniques have been devised to process CSLDV measurements from structures under controlled input. The authors recently extended CSLDV to the case where the input forces are unmeasured random white noise, using the harmonic power spectrum of a time periodic system. The harmonic power spectrum is analogous to the power spectrum used in the identification of time invariant systems, but with many additional harmonics for each mode, requiring an additional effort in modal parameter estimation. This paper presents a variant on the harmonic power spectrum, proposing a simplified algorithm based on the lifting approach. Lifting causes all sideband peaks in the harmonic power spectrum to collapse into a single peak in the range from zero to half of the scan frequency, so the spectra are far easier to interpret. The proposed algorithm is first evaluated on a simulated beam, and found to give results that are comparable with those obtained by the harmonic power spectrum method, yet the data reduction with the lifting approach is much simpler. This algorithm is then employed to identify the first several modes of a parked wind turbine under wind excitation, using a new long range remote sensing vibrometer. The speckle noise is found to be remarkably small even at a standoff distance of 77m and a surface scan velocity of 500m/s without any surface treatment.

A Wavelet Approach for Identification of Linear Time Invariant System

Wavelet transformation has been applied to various problems of system identification. In this paper, a wavelet based approach for the identification of time-invariant system is proposed. In this approach, mother wavelet is used for excitation to find the impulse response, which can be estimated at half the available number of points of the sampled output sequence. This method has been compared with some other standard techniques such as frequency chirp, coherence function and inverse filtering. In chirp method, wideband excitation such as frequency chirp is used. Frequency response is obtained as the DFT of the output of the system for time-domain input. Inverse method uses SVD function to find pseudoinverse. Coherence function has been used to identify the system using MATLAB function tfestimate. The performances of the methods are demonstrated by means of experimental investigation.

Prefiltering of Frequency Domain Data Based on Wold's Decomposition for System Identification

In this paper, we propose a prefiltering method of frequency domain data for linear multiple-input multiple-output time-invariant system identification. This paper only deals with the case where the frequency domain input-output data is given by discrete Fourier transform. The concept of the proposed method is to eliminate the disturbance factor from the output data, it can be performed by orthogonal decomposition based on Wold's decomposition. Thus, by employing MIMO system identification method based on the frequency domain data given by the discrete Fourier transform, the data implemented by the proposed method can derive more accurate model. LQ decomposition is the key tool in the proposed method, it actually computes the orthogonal decomposition of the frequency domain data instead of Wold's decomposition. The efficacy of the proposed method is shown in numerical experimental.

Toeplitz Compressed Sensing Matrices With Applications to Sparse Channel Estimation

Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of high-dimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zero-mean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, time-invariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that time-domain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional least-squares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies.

Subspace closed loop identification using the integration of MOESP and N4SID methods

Linear identification of time invariant systems operating in closed loop is of special interest for a large number of engineering applications. There are different techniques and methods to carry out this type of identification. For example, modifying the N4SID method, one can derive a closed loop subspace identification method. The same can be done using the MOESP method. Based on them, the MON4SID method is introduced, which estimates the extended observability matrix and the state sequence directly from a LQ decomposition, using a combination of the techniques contained in both, MOESP and N4SID. This new method uses an algorithm to identify state space model of a plant in a closed loop system, in the same way as in MOESP method. The advantage of the proposed algorithm is that it does not require any knowledge about the controller, whereas such information is essential for other methods (e.g. N4SID). The disadvantage of this algorithm is that it needs a great amount of data to obtain better estimates. A simulated process to show the performance of this algorithm is presented.

Frequency Domain Identification of Linear, Deterministically Time-Varying Systems

This paper proposes an extension of the framework of linear time invariant system identification to linear slowly time-varying system identification. The class of systems considered is described by ordinary differential equations with coefficients varying piecewise polynomially with time. The estimation is performed in the frequency domain using multisine excitation signals within an errors-in-variables framework. It is also discussed how multisine excitations provide estimates of, on the one hand, the speed of variation of the system and, on the other hand, an estimation of a non-parametric noise model.

Non-parametric linear time-invariant system identification by discrete wavelet transforms

We describe the use of the discrete wavelet transform (DWT) for non-parametric linear time-invariant system identification. Identification is achieved by using a test excitation to the system under test (SUT) that also acts as the analyzing function for the DWT of the SUT's output, so as to recover the impulse response. The method uses as excitation any signal that gives an orthogonal inner product in the DWT at some step size (that cannot be 1). We favor wavelet scaling coefficients as excitations, with a step size of 2. However, the system impulse or frequency response can then only be estimated at half the available number of points of the sampled output sequence, introducing a multirate problem that means we have to 'oversample' the SUT output. The method has several advantages over existing techniques, e.g., it uses a simple, easy to generate excitation, and avoids the singularity problems and the (unbounded) accumulation of round-off errors that can occur with standard techniques. In extensive simulations, identification of a variety of finite and infinite impulse response systems is shown to be considerably better than with conventional system identification methods.

Higher-order statistics for Rice's representation of cyclostationary signals

The role played by the higher-order wide-sense cyclostationarity properties in Rice's representation of a signal is investigated. Specifically, formulas relating higher-order cyclic spectra of signals involved in Rice's representation of a signal exhibiting cyclostationarity are derived. It is shown that some known results relative to second-order cyclic spectra cannot be extended to higher orders. Furthermore, with reference to strictly band-limited signals, inclusion relationships for the spectral supports (in the space of the cycle and spectral frequencies) of the considered higher-order spectra are determined. Finally, as examples of application, the problem of cycloergodicity for the higher-order cyclic spectra of a quadrature amplitude modulated signal and the bandpass linear time-invariant system identification are considered.

Wavelet-based linear system modeling and adaptive filtering

It is shown how linear time-varying systems can be modeled in several different ways by discrete-time wavelets or, more generally, by some set of functions. Interpretation of physical meanings, possible efficiency, and other characteristics of the modeling are considered. System identification minimizing the mean square output error is studied. Optimal coefficients and the corresponding minimum mean square error are found, and they are, in general, time varying. Least-mean-square adaptive filtering algorithms are derived for on-line filtering and system Identification. Theoretically and by simulations, the advantages of using wavelet-based filtering are shown: separation of adaptation effects from unknown time-varying system behavior and fast convergence. Adaptive coefficients estimated by a recursive-least-square algorithm can tend toward constants, even in the case of time-varying systems. Time-invariant system identification and adaptive filtering is given as a special case of the general time-varying setting.

Micro-Robotics and the Study of Muscle: Special Problems in Control, System Identification and Modelling

In order to study nature's micro-machine, the muscle cell, we have developed, MR-1, a special purpose micro-motion robot. MR-1 has two limbs which are able to grasp single living muscle cells at each end and perform mechanical and optical experiments on them. An interesting set of system identification, control and modelling problems arise in this work. Traditional approaches to controlling the micro-robot have failed to meet our requirements and in our muscle studies, linear time-invariant system identification methods have not yielded useful scientific insights. In this paper the background to these problems is given together with our attempts to provide solutions to them. In some cases our attempts have been fruitful in others we have only succeeded in defining the problem and so must await future work.

Application of Generalized Orthogonal Polynomials to Parameter Estimation of Time-Invariant and Bilinear Systems

Generalized orthogonal polynomials (GOP) which can represent all types of orthogonal polynomials and nonorthogonal Taylor series are first introduced to estimate the parameters of a dynamic state equation. The integration operation matrix (IOP) and the differentiation operation matrix (DOP) of the GOP are derived. The key idea of deriving IOP and DOP of these polynomials is that any orthogonal polynomial can be expressed by a power series and vice versa. By employing the IOP approach to the identification of time invariant systems, algorithms for computation which are effective, simple and straightforward compared to other orthogonal polynomial approximations can be obtained. The main advantage of using the differentiation operation matrix is that the parameter estimation can be carried out starting at an arbitrary time of interest. In addition, the computational algorithm is even simpler than that of the integral operation matrix. Illustrative examples for using IOP and DOP approaches are given. Very satisfactory results are obtained.

A generalized cubic spline technique for identification of multivariable systems

The application of cubic splines to the identification of time-invariant systems is considered. The use of splines, initially proposed by Bellman in 1971, has been extended to the multidimensional case. In addition, the effects of noise on the identification procedure are considered and techniques are presented for improving the identification accuracy. A general spline technique, used in conjunction with a Kalman estimation procedure, has been developed for identifying physical systems described by a set of first-order differential equations. This method has been found to be superior to the exponential fitting technique proposed by Prony and to other finite-difference methods.

A method for discrete system identification and approximation

This paper considers a problem of time invariant system identification where the system and the signals have a discrete representation in the z transform domain. The problem is considered for three types of inputs; deterministic, random, and mixed. The solution shows how, under certain conditions, it is possible to identify unknown linear systems with discrete representations. Also under consideration is the discrete time domain approximation problem known as the compensation problem. It is further shown that all of the results that apply to discrete systems with one dimensional representation can be extended to systems with N dimensional representations. The solutions are determined by triangular matrix equations, which avoid tedious matrix inversions and lend themselves to implementation with existing digital computer equipment.